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- 1. Let T:V rightarrow W be a linear transformation, and let v1, v2, ..., vk denote vectors in V. If {T(v1), ..., T(vk)} is independent, show that {v1, ..., vk} is independent. If {v1, v2, ... ,vk} spans V, show that {T(v1), T(v2), ... ,T(vk)} spans im T. Solution a) Suppose the v i 's were dependent. Then there are some (not all zero constants) such that c 1 v 1 +c 2 v 2 +...+c k v k =0 But since T is a linear transformation T(0)=0 hence 0 = T(c 1 v 1 +c 2 v 2 +...+c k v k ) = c 1 T(v 1 )+c 2 T(v 2 )+...+c k T(v k ) . But since the c i 's are not all zero, this means that T(v i ) are linearly dependent. (contradiction) Su the vi's must all be independent. b) let w be any vector in im T. Then there must be some vector v such that w=Tv. But since v i spans V, there are some constants such that c 1 v 1 +c 2 v 2 +...+c k v k =v. therefore w=Tv=c 1 T(v 1 )+c 2 T(v 2 )+...+c k T(v k ). Therefore T(v i ) span im T.